PLAYGROUND

DISTRIBUTIONS / CONTINUOUS / GENERALIZED PARETO

Generalized Pareto distribution

What it represents

The Generalized Pareto distribution, abbreviated GPD, models exceedances above a high threshold. Its c parameter summarizes the tail regime and sigma calibrates excess magnitude.

Historical clue

The modern extreme-value connection comes from Pickands and independently Balkema–de Haan: under broad conditions, scaled threshold excesses converge to a GPD.

Relationships that clarify its use

It is GEV’s companion: GEV for block maxima, GPD for threshold exceedances. For c>0 it approaches Pareto tail behaviour; for c=0 it is shifted exponential.

Data examples

  • river flow, rainfall, wind, or pollution above a threshold
  • financial losses, large claims, and operational risk

Modelling warning

Fits can be highly threshold- and dependence-sensitive; do not extrapolate extreme quantiles merely because the local GPD fit looks good.

What remains after crossing a high threshold

Pickands, Balkema, and de Haan established that for a broad class of populations, excesses over sufficiently high thresholds approach a Generalized Pareto law. This is the peaks-over-threshold method, an alternative to retaining only one maximum per block.

The word sufficiently is the practical problem. A low threshold creates bias by admitting nonextreme data; a very high one leaves few excesses and high variance. Stability of xi, mean excess plots, and process knowledge all help. Excesses should also be approximately independent. In storms or transactions, several nearby values may belong to one episode and require declustering before return periods can be interpreted.

Decision guide

A good candidate when: excesses above a high threshold are analysed and the main interest is in extreme probabilities or quantiles.

Compare it with: GEV for block maxima. Check shape-parameter stability across thresholds and temporal dependence among exceedances before extrapolating.

References

  • SciPy reference: scipy.stats.genpareto — definition and parameterization

  • Balkema, A. A. & de Haan, L. (1974). Residual life time at great age. The Annals of Probability, 2(5), 792–804.

  • Johnson, N. L., Kotz, S. & Balakrishnan, N. (1995). Continuous Univariate Distributions, 2nd ed., Vol. 2. Wiley.

  • Pickands, J. (1975). Statistical inference using extreme order statistics. The Annals of Statistics, 3(1), 119–131.

  • Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.

Generalized Pareto Distribution: equations and calculator

Distribution defintion

X∼GeneralizedPareto(c,μ,σ)X\sim\mathrm{GeneralizedPareto}\left(c,\mu,\sigma\right)

Distribution domain

if c⩾0: x∈(μ,∞),if c<0: x∈(−∞,μ−σc)\text{if }c\geqslant 0:\ x\in\left(\mu,\infty\right),\quad \text{if }c<0:\ x\in\left(-\infty,\mu-\frac{\sigma}{c}\right)

Parameters domain and parameters constraints

c∈R,μ∈R,σ∈R+c\in\mathbb{R},\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=1−(1+cz(x))−1/cF_{X}\left(x\right)=1-(1+c z(x))^{-1/c}

Probability density function

fX(x)=1σ(1+cz(x))−(1/c+1)f_{X}\left(x\right)=\frac{1}{\sigma}(1+c z(x))^{-(1/c +1)}

Percent point function/Sample

FX−1(u)=μ+σ(u−c−1)cF^{-1}_{X}\left(u\right)=\mu+\frac{\sigma (u^{-c}-1)}{c}

Non-central parametric moments

μk′=E[Xk]=∫−∞∞xkfX(x)dx=(−1)kck∑i=0k(ki)(−1)i1−ciif <1k\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\frac{\left(-1\right)^{k}}{c^{k}}\sum_{i=0}^{k}\binom{k}{i}\frac{\left(-1\right)^{i}}{1-ci}\quad \text{if }<\frac{1}{k}

Parametric mean

Mean(X)=μ+σμ1′=μ+σ1−cif c<1\mathrm{Mean}(X)=\mu+\sigma\mu'_{1}=\mu+\frac{\sigma}{1-c} \quad \text{if } c<1

Parametric variance

Variance(X)=σ2(μ2′−μ1′2)=σ2(1−c)2(1−2c)if c<1/2\mathrm{Variance}(X)=\sigma^2(\mu'_{2}-\mu'^{2}_{1})=\frac{\sigma^2}{(1-c)^2(1-2c)} \quad \text{if } c<1/2

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=2(1+c)1−2c(1−3c)if c<1/3\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{2(1+c)\sqrt{1-2c}}{(1-3c)} \quad \text{if } c<1/3

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=3(1−2c)(2c2+c+3)(1−3c)(1−4c)if c<1/4\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{3(1-2c)(2c^2+c+3)}{(1-3c)(1-4c)} \quad \text{if } c<1/4

Parametric median

Median(X)=μ\mathrm{Median}(X)=\mu

Parametric mode

Mode(X)=μ+σ(2c−1)c\mathrm{Mode}(X)=\mu+\frac{\sigma( 2^{c} -1)}{c}

Additional information and definitions

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}